Recently, a number of theoretical investigations have been devoted to the water wave problem with nonzero vorticity due to
its relevance to oceanography and coastal engineering where the influence of currents on waves may play an important role [1]}, [2]}.
Unlike the irrotational case, a full-dimensional computation is in general required to solve for the vorticity field.
This has prompted efforts to propose simplified models for rotational waves in the long-wave shallow-water regime [3]}, [4]}.
Of special interest is the case of nonzero constant vorticity which corresponds to vertically shear flow with a linear profile.
The direction of the underlying current can be that of wave propagation (co-propagating) or opposite (counter-propagating).
Similar to the irrotational water wave problem, this particular case allows for a lower-dimensional reformulation of the governing equations.
As a consequence, it has received much attention in both mathematical and numerical studies,
regarding e.g. the existence and stability of steadily progressing wave solutions [5]}, [6]}, [7]}, [8]}, [9]}, [10]},
the flow structure beneath waves [11]}, or the focusing of transient waves by an adverse current [12]}, [13]}.
